We consider the enstrophy cascade in forced two-dimensional turbulence with a linear drag force. In the presence of linear drag, the energy wavenumber spectrum drops with a power law faster than in the case without drag, and the vorticity field becomes intermittent, as shown by the anomalous scaling of the vorticity structure functions. Using previous theory, we compare numerical simulation results with predictions for the power law exponent of the energy wavenumber spectrum and the scaling exponents of the vorticity structure functions ζ2q. We also study, both by numerical experiment and theoretical analysis, the multifractal structure of the viscous enstrophy dissipation in terms of its Rényi dimension spectrum Dq and singularity spectrum f (α). We derive a relation between Dq and ζ2q, and discuss its relevance to a version of the refined similarity hypothesis. In addition, we obtain and compare theoretically and numerically derived results for the dependence on separation r of the probability distribution of δ r ω, the difference between the vorticity at two points separated by a distance r. Our numerical simulations are done on a 4096 × 4096 grid.

The time asymptotic decay of the variance of a passive scalar in a chaotic flow is studied. Two mechanisms for this decay, which involve processes at short and long length scales, respectively, are considered. The validity of the short length scale mechanism, which is based on Lagrangian stretching theory, is discussed. We also investigate the regimes of applicability and observable signatures of the two mechanisms. Supporting evidence is provided by high resolution numerical experiments.

New analytic estimates of the rate at which parametric subharmonic instability (PSI) transfers energy to high-vertical-wavenumber near-inertial oscillations are presented. These results are obtained by a heuristic argument which provides insight into the physical mechanism of PSI, and also by a systematic application of the method of multiple time scales to the Boussinesq equations linearized about a ‘pump wave’ whose frequency is close to twice the inertial frequency. The multiple-scale approach yields an amplitude equation describing how the 2f0-pump energizes a vertical continuum of near-inertial oscillations. The amplitude equation is solved using two models for the 2f0-pump: (i) an infinite plane internal wave in a medium with uniform buoyancy frequency; (ii) a vertical mode one internal tidal wavetrain in a realistically stratified and bounded ocean. In case (i) analytic expressions for the growth rate of PSI are obtained and validated by a successful comparison with numerical solutions of the full Boussinesq equations. In case (ii), numerical solutions of the amplitude equation indicate that the near-inertial disturbances generated by PSI are concentrated below the base of the mixed layer where the velocity of the pump wave train is largest. Based on these examples we conclude that the e-folding time of PSI in oceanic conditions is of the order of ten days or less.

We examine the basic properties and stability of isolated vortices having uniform potential vorticity in a non-hydrostatic rotating stratified fluid, under the Boussinesq approximation. For simplicity, we consider a uniform background rotation and a linear basic-state stratification for which both the Coriolis and buoyancy frequencies, f and N , are constant. Moreover, we take f /N ≪ 1, as typically observed in the Earth's atmosphere and oceans. In the small Rossby number 'quasigeostrophic' limit, when the flow is weak compared to the background rotation, there exists exact solutions for steadily-rotating ellipsoidal volumes of uniform potential vorticity in an unbounded flow (Zhmur & Shchepetkin 1991;Meacham 1992). Furthermore, a wide range of these solutions are stable so long as the horizontal and vertical aspect ratios λ and µ do not depart greatly from unity (Dritschel et al. 2005). In the present study, we examine the behaviour of ellipsoidal vortices at Rossby numbers up to near unity in magnitude. We find that there is a monotonic increase in stability as one varies the Rossby number from nearly −1 (anticyclone) to nearly +1 (cyclone). That is, quasi-geostrophic vortices are more stable than anticyclones at finite (negative) Rossby number, and generally less stable than cyclones at finite (positive) Rossby number. Ageostrophic effects strengthen both the rotation and the stratification within a cyclone, enhancing its stability. The converse is true for an anticyclone. For all Rossby numbers, stability is reinforced by increasing λ towards unity or decreasing µ. An unstable vortex often restabilises by developing a near-circular cross section, typically resulting in a roughly ellipsoidal vortex, but occasionally a binary system is formed. Throughout the nonlinear evolution of a vortex, the emission of inertiagravity waves is negligible across the entire parameter space investigated. Thus, vortices at small to moderate Rossby numbers, and any associated instabilities are (ageostrophically) balanced. A manifestation of this balance is that, at finite Rossby number, an anticyclone rotates faster than a cyclone.